Lecture 5

# Lecture 5
## The Effects Model
### Psych 10 C
### University of California, Irvine
### 04/08/2022

---

## Null Model

- Last class we used mathematical notation and the Normal distribution to 
represent what we called the **Null Model**

- The Null model assumed that observations from every participant `$i=1,\dots,I$` 
are sampled from the same distribution, regardless of their group `$j=1,2$`.

- The Null model formalized an initial verbal hypothesis: that there are **no 
differences between groups**.

- The **Null Model** is defined as: `$$y_{ij} \sim \text{Normal}(\mu, \sigma^2)$$`

- Finally, we said that since we don't know the values of the parameters `$\mu$` 
and `$\sigma^2$` that fully define the Normal distribution, we needed to infer 
(learn) them from our observations.

- This can be done through a process called **Statistical Inference**.

---

## Statistical Inference for the Normal distribution

- One of the key advantages of the Normal distribution is that its parameters 
can be tracked directly to the Expectation and Variance of the random variable.

- In other words, the Null model assumes that our observations are random 
variables that follow a Normal distribution with parameters `$\mu$` and `$\sigma^2$`.

- If a random variable `$y$` follows a Normal distribution with parameters `$\mu$` 
and `$\sigma^2$`, then we know that the following two statements are **TRUE**:
`$$\mathbb{E}(y) = \mu$$` and `$$\mathbb{V}ar(y) = \sigma^2$$`

- Remember that we already have a very good approximation for both `$\mathbb{E}(y)$`
and to the `$\mathbb{V}ar(y)$`!

---

# Estimators

---

## Estimators

- In Week 1 we talked about statistics as functions of our observations.

- When a statistic is used to approximate a parameter (e.g. `$\mu$`) in a 
statistical model, it is called an **estimator**.

- Since we know that our best statistic for the expected value of a r.v. is
the mean (i.e. the average of our observations), we can use it as an 
**estimator** for `$\mu$`.

- We represent these estimators by adding a "hat" on top of the Greek character
corresponding to the parameter they estimate: 
`$$\hat{\mu} = \frac{1}{n} \sum_{j} \sum_{i} y_{ij}$$`

- Here, `$n$` represents the total number of observations that we add together to 
calculate the mean, while the indices `$i$` and `$j$` denote the observation number 
and the group, respectively.

---

## Variance estimator

- We also said that a good approximation for the variance of a r.v. is the 
sample variance `$s^2$`. However, this time we will write it slightly different
to make other models easier to understand.

- Our estimator for the variance will be denoted as: 
`$$\hat{\sigma}^2_0 = \frac{1}{n} \sum_j \sum_i \left(y_{ij}-\hat{\mu}\right)^2$$`

- If you look at your previous notes, you will see that we replaced the sample 
mean `$\bar{y}$` with our **estimator** `$\hat{\mu}$`, and that now we refer to it 
as `$\hat{\sigma}_0^2$` rather than `$s^2$`.

- The subscript "0" indicates that this estimator corresponds to the variance of 
the **Null Model**.

---

## The Null Model

- In the case of our Null model, we can use our new estimators `$\hat{\mu}$` 
and `$\hat{\sigma}_0^2$` to refer to:

1. The Model Prediction: `$\hat{\mu}$`
  
--

1. The Mean Squared Error: `$\hat{\sigma}_0^2$`
  
--

- The variance estimator `$\hat{\sigma}_0^2$` is referred to as the Mean Squared 
Error because it captures the average distance (in squared units) between what 
we expect to observe ( `$\hat{\mu}$` ) and what we actually observed ( `$y_{ij}$` ).

- When using this type of statistical models (as we'll do for most of this 
class), we will also be interested on the Sum of Squared Errors, which is 
denoted as: 
`$$SSE_0 = \sum_j \sum_i \left(y_{ij}-\hat{\mu}\right)^2$$`

- Again, we add "0" as a subscript to make it clear that this Sum of Squared 
Errors is associated to the **Null Model**.

---

- Form teams of 3 and calculate the model prediction, the sum of squared errors 
and the mean squared error that would correspond to the **Null Model** for the 
smokers data:

<div id="htmlwidget-ec61b4f418fbae925c09" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-ec61b4f418fbae925c09">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["1","2","3","4","5","6","7","8"],[1,2,3,4,5,6,7,8],[78.3,80,73.6,77.4,68,64.4,70.8,65.5],["non_smoker","non_smoker","non_smoker","non_smoker","smoker","smoker","smoker","smoker"]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th> <\/th>\n      <th>id<\/th>\n      <th>lung_capacity<\/th>\n      <th>smoke_status<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"pageLength":8,"columnDefs":[{"className":"dt-right","targets":[1,2]},{"orderable":false,"targets":0}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[8,10,25,50,100]}},"evals":[],"jsHooks":[]}</script>

---

## Smokers data: Null Model

.can-edit.key-likes[
- Prediction: 
]

.can-edit.key-likes[
- SSE: 
]

.can-edit.key-likes[
- mean Squared Error: 
]

---

# The Effects Model

---

## Effects Model

- Now we will formalize our second model.

- Our original problem stated that we wanted to know if there were any 
differences in lung capacity between smokers and non-smokers.

- Our first model was constructed from the idea that there are no differences 
between groups.

- We will call the model that assumes that **groups are different** the 
**Effects Model**.

---

## Effects Model

- Following the logic we used to formalize the Null Model, we will keep on using
our observations' notation `$y_{ij}$` and the Normal distribution as the basis for
this second statistical model.

- In mathematical notation, the **Effects Model** would be: 
`$$y_{i1}\sim\text{Normal}(\mu_1,\sigma_e^2)$$` `$$y_{i2}\sim\text{Normal}(\mu_2,\sigma_e^2)$$`

or `$$y_{ij}\sim\text{Normal}(\mu_j,\sigma_e^2)$$`

- Both ways of writing the model convey the same information, but the second 
one is just shorter.

- This new statistical model formalizes the idea that the `$i = 1, \dots, 4$` 
observations on each group `$j = 1, 2$` are sampled from two different Normal 
distributions, one centered at `$\mu_1$` and the other one centered at `$\mu_2$`.

---

## Effects Model

- Another important thing to notice from our definition of the model is that we 
assume that both distributions have the same variance, ($\sigma_e^2$).

- This means that our model assumes that the only difference between the 
two groups is their expected value and not how much they vary.

- We can now use all of our observations to calculate a single error estimate 
for this model.

- Why is it called the **Effects Model**?

- The difference between `$\mu_2$` and `$\mu_1$` can be interpreted as the effect
on our random variable that comes from being in group 2 in comparison to being 
in group 1.

- In our smoking example, the difference between `$\mu_2$` and `$\mu_1$` can be 
interpreted as the effect that smoking has on lung capacity.

- In other words, how much do we *expect* lung capacity to change if a person 
smokes in comparison to a person that does not.

---

# Graphical Representation Effects Model

- As we did with the Null Model, we can also represents the effects model 
graphically.

- Notice that we still don't have values for our parameters, so we can make a 
graph that shows that we expect each group to be represented by a 
different normal distribution with a different mean but same variance.

- We can use the following R code to generate that representation:

```r
par(mai = c(1,0.1,0.1,0.1))
curve(dnorm(x, mean = 0, sd = 1), from = -4, to = 6,
      axes = FALSE, ann = FALSE, col = "red", lwd = 3)
curve(dnorm(x, mean = 2, sd = 1), col = "blue", add = T, lty = 3, lwd = 3)
box(bty = "l")
mtext(text = "Lung capacity", side = 1, line = 2, cex = 1.6)
legend("topleft", bty = "n", col = c("blue","red"), 
       legend = c("non-smokers", "smokers"), lty = c(3, 1))
```
]

.pull-right[
<img src="data:image/png;base64,#lec-5_files/figure-html/normal-effect-out-1.png" style="display: block; margin: auto;" />
]

---

## Graphical Representation

- Notice that in the previous plot we do not have values assigned to the 
variable lung capacity.

- This is because we only want to show that the model is assuming that the 
groups are samples from two different normal distributions.

- However, we can (and should) formalize our Theories about the world before
even looking at the data.

- In other words, we will always have a specification of our models even before 
running an experiment (so we can't estimate the values of our parameters).

---

## Estimators for the Effects model

- Now that we have specified our new model (that assumes that the groups are 
different), we have 3 parameters for which we want to find appropriate 
estimators.

- Our parameters are `$\mu_1$`, `$\mu_2$` and `$\sigma_e^2$`. Again, our estimators 
will be a statistic (a function of the sample) and we will denote them with a 
hat.

`$$\hat{\mu_1} = \frac{1}{n_1} \sum_i y_{i1}$$`

`$$\hat{\mu_2} = \frac{1}{n_2} \sum_i y_{i2}$$`

and finally,

`$$\hat{\sigma}_e^2 = \frac{1}{n} \sum_j \sum_i (y_{ij} - \hat{\mu}_j)^2$$`

- Where `$n$` represents the total number of observations, and `$n_j$` represents 
the number of observations in group `$j = 1, 2$`

---

## Estimators for the Effects Model

- Notice that our estimators for `$\mu_1$` and `$\mu_2$` are just the mean of our
observations by group!

- Again, we can interpret `$\hat{\mu}_j$` as the prediction of the model for each
observation `$i = 1, \dots, 4$` of group `$j = 1, 2$`.

- Then we can interpret our estimator for `$\hat{\sigma}_e^2$` as the average 
error in the predictions of the model. In other words, how far on average are 
our observations from our best prediction `$\hat{\mu}_j$` of the group.

- As we did with the null model, we will be interested again in the total error
of the model, also known as the Sum of Squared Error of the **Effects Model**.

- We will denote the Sum of Squared Error of the **Effects Model** as:
`$$SSE_e = \sum_j \sum_i (y_{ij} - \hat{\mu}_j)^2$$`

---

- Form teams of 3 and calculate the model predictions for each group, the sum of 
squared errors and the mean squared error of the **Effects Model** for the 
smokers data:

<div id="htmlwidget-c82de8d69bb2fca99b29" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-c82de8d69bb2fca99b29">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["1","2","3","4","5","6","7","8"],[1,2,3,4,5,6,7,8],[78.3,80,73.6,77.4,68,64.4,70.8,65.5],["non_smoker","non_smoker","non_smoker","non_smoker","smoker","smoker","smoker","smoker"]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th> <\/th>\n      <th>id<\/th>\n      <th>lung_capacity<\/th>\n      <th>smoke_status<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"pageLength":8,"columnDefs":[{"className":"dt-right","targets":[1,2]},{"orderable":false,"targets":0}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[8,10,25,50,100]}},"evals":[],"jsHooks":[]}</script>

---

## Smokers data: Effects Model

.can-edit.key-likes[
- Prediction: 
]

.can-edit.key-likes[
- SSE: 
]

.can-edit.key-likes[
- mean Squared Error: 
]